In a small saucepan, bring the water to a boil over medium-high heat.
Whisk in the chickpea flour and salt. Cook for 1 minute, whisking constantly. Reduce the heat to medium-low; cook, whisking, for 4 to 5 minutes longer, until thickened. Transfer mixture to a small, heatproof bowl; cover and refrigerate until cold.
Transfer the cold chickpea mixture to a food processor or blender. Add the vegetable broth, lemon juice, mustard, and cayenne; process until blended and very creamy. Transfer to a sealable jar or another covered container.
Store in the refrigerator for up to 3 months. Use as you would any other mayonnaise or aioli!
Lemon Mayonnaise: Increase the lemon juice to 2 tablespoons and add 1-1/2 teaspoons of finely grated lemon zest when blending all of the ingredients in a blender.